Answer
$67$
Work Step by Step
Step 1. Assuming the total number of people is $x, 50\leq x\leq 80$, the total cost will be:
$cost=6000+32x$
Step 2. The total income will be:
$income=xp$
where $p$ is the price per person and
$p=200-2(x-50)=300-2x$
Step 3. The total profit will be
$P=x(300-2x)-6000-32x=268x-2x^2-6000$.
Take its derivative to get $P'=268-4x$ and let $P'=0$ to get $x=67$ .
Step 4. Check $P''=-4\lt0$, indicating a concave down region with a maximum. Thus the above $x$ value will give a maximum profit.
